Fourier-Hermite methods for perturbed harmonic oscillator problems

Abstract

In this work, we address the problem of solving non-linear Schrödinger equations numerically and find that an exact particular splitting technique shows superior performance to commonly used methods. We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a harmonic potential or a small perturbation to it. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using Fast Fourier Transforms. To split the system into the quantum harmonic oscillator problem and the remaining part allows to get higher accuracies in many cases, but it requires to change between Hermite basis functions and the coordinate space. We show how to build new methods which combine the advantages of using Fourier methods while solving the harmonic oscillator exactly.